Book edition 9th Edition

Author(s) Raymond A. Serway, John W. Jewett

Pages 1624 pages

ISBN 9781133947271

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Short Answer

A chain of length L and total mass M is released from rest with its lower end just touching the top of a table as shown in Figure P9.96a. Find the force exerted by the table on the chain after the chain has fallen through a distance x as shown in Figure P9.96b. (Assume each link comes to rest the instant it reaches the table.

The total force exerted by the table on chain during falling distance x is $\frac{3 M g x}{L}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identification of given data

The mass of chain is $M$

The length of chain is $L$

The distance fallen by the chain through hole is $x$

Step 2: Linear Mass Density

The linear mass density of an object is the mass of the unit length of the object. it is given as –

$λ = \frac{M}{L}$

Step 3: Determination of force exerted by table on the chain

The mass of small element of chain is given as:

$d m = λ \cdot d x = (\frac{M}{L}) d x$

The weight of chain length fallen is given as:

$d w = λ g x = (\frac{M}{L}) g x$

The speed of the chain after falling distance x is given as:

$v = \sqrt{2 g x}$

The force exerted on chain after falling distance x is given as:

$F = v (\frac{d m}{d t})$

For the given values, the above equation becomes-

$F = v (\frac{((\frac{M}{L}) d x)}{d t}) = (\frac{M}{L}) v (\frac{d x}{d t}) = (\frac{M}{L}) v (v) = (\frac{M}{L}) v^{2}$

solving further,

$F = (\frac{M}{L}) {(\sqrt{2 g x})}^{2} F = \frac{2 M g x}{L}$

The total force exerted by the table on chain during falling distance x is given as:

$F_{t} = F + w$

For the given values, the above equation becomes-

$F_{t} = (\frac{2 M g x}{L}) + (\frac{M g x}{L}) = \frac{3 M g x}{L}$

Therefore, the total force exerted by table on chain during falling distance x is $\frac{3 M g x}{L}$ .

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Physics For Scientists & Engineers

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Short Answer

Step by Step Solution

Step 1: Identification of given data

Step 2: Linear Mass Density

Step 3: Determination of force exerted by table on the chain

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Physics For Scientists & Engineers

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Short Answer

Step by Step Solution

Step 1: Identification of given data

Step 2: Linear Mass Density

Step 3: Determination of force exerted by table on the chain

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